Final answer:
To find the critical numbers of the function and the open intervals where the function is increasing or decreasing, we need to find the derivative and analyze its sign. Applying the First Derivative Test, we can identify the relative extremum.
Step-by-step explanation:
To find the critical numbers of the function f(x) = 2x^3 + 9x^2 - 24x, we first need to find the derivative f'(x) and set it equal to zero. Taking the derivative using the power rule, we get f'(x) = 6x^2 + 18x - 24. Setting this equal to zero and solving for x, we get x = -4 and x = 1. Therefore, the critical numbers of f are -4 and 1.
To find the open intervals on which the function is increasing or decreasing, we need to examine the sign of the derivative. We can do this by considering the values of x within and outside the critical interval. Testing points in these intervals, we find that f'(x) > 0 for x < -4, -4 < x < 1, and x > 1. Therefore, the function is increasing on the intervals (-∞, -4) and (1, ∞) and decreasing on the interval (-4, 1).
To apply the First Derivative Test and identify the relative extremum, we need to check the sign changes in the derivative. Since f'(x) changes from positive to negative at x = -4, we have a relative maximum at x = -4. There is no sign change at x = 1, so there is no relative minimum at x = 1.